Starting again from \((w,v) = (1/4,2)\) , iterates are initially attracted to \(S_{-,\varepsilon }^a\)  and subsequently track iterates of the map (REF ) towards the fold \(F_-\) , similarly to cases I, II and III. After leaving a neighbourhood of the fold, complicated dynamics corresponding to a slow (leftward) drift through part of a period-doubling cascade occurs. The details of this transition are beyond the scope of this work, however the reader is referred to [1]} for details on the period-doubling cascade in the corresponding layer problem, and to [2]} for details on the slow drift through a period-doubling cascade in a particular `normal form'. Once iterates are close enough to \(w_{flip}\) , the dynamics becomes non-chaotic and orientation-reversing as in case III, and eventually approach a neighbourhood of \(S_{+,\varepsilon }^a\)  after a delay effect (again described by [2]}). The subsequent tracking of iterates along \(S_{+,\varepsilon }^a\)  and the observed return to \(S_{-,\varepsilon }^a\)  is similar to that described for cases II and III. Assuming a simple-jump like behaviour near the folds \(F_\pm \) , this geometric sequence repeats itself and provides a mechanism for chaotic bursting behaviour.
